442 XV NORMED ALGEBRAS AND SPECTRAL THEORY is such that D = P~D = DP+, and is therefore of rank m (note that, if y e E and Xl = (Al + i/)-1 - y, then Xj. e dom(jET*) and y = (H* + il) Xj.) 14. A closed unbounded hermitian operator H on E is said to be major ized (resp. minorized) if there exists a real number c such that (H x \ x) <: c(x \ x) (resp. (H - x | x) ^ c(# | #)) for all # e dom(ff). The operator H is said to be positive if (H x | x) ^ 0 for all x e dom(jy). Thus jfiT is majorized (resp. minorized) if and only if there exists c e R such that cl H (resp. H cl) is positive. (a) Let .ftT be a positive closed hermitian operator on E. The hermitian form f(xt y) (x + H-x\y) on dom(H) makes it a prehilbert space, which may be con- sidered as a dense subspace of a Hilbert space G. Show that the canonical injection j of domC/HO into E extends by continuity to an injection of G into E, so that G may be identified with a subspace of E. We denote the scalar product on G again by/(x, y). Show that there exists a continuous linear mapping B of E into G (where G is con- sidered as a Hilbert space) such that (x \y) =/(.# Jt, y) for all xeE and y e G. Considered as a mapping of E into E, the operator 3 is positive, continuous, self- adjoint and of norm <|1; we have (B-x\x)^ \\B-x\\2, and B(I + H ) - y = y for all y e dom(AT). Deduce that A B""1 /is an unbounded positive self-adjoint operator, such that dom(/4) 5(E), which extends If. Hence the defects of £Tare equal. (b) Suppose that the defects of H are equal and finite, say m. If A i is any self-adjoint extension of H, show that the intersection of Sp(^i) and the interval ] oo, 0[ consists of eigenvalues of Ai9 and that the total number of such eigenvalues, each counted according to its multiplicity, is ^m. (For each compact interval J c ] oo, 0[, show that the image of E under the projector (pj(Ai) (Section 15.12, Problem 7), which is contained in dom(Ai), cannot have dimension >m, otherwise it would contain a nonzero vector x<=dom(H) (Problem 13(a)); show that this conclusion would con- tradict the positivity of H.) 15. Let H be a closed hermitian operator on E, with defect (m, n\ so that H has defect (n, m). Show that H is the restriction to E of an unbounded self-adjoint operator A on the Hilbert sum E© E. 16. (a) Let G be an analytic function on the disk B : \z\ < 1, such that ^G(z) ^ 0 in this disk. Show that the function K(ii, v) = (G(«) + GW)/(1 - irf) on B x B is of positive type (Section 6.3, Problem 4). (Use Section 14.11, Problem 18.) (b) Let F be an analytic function on the half-plane D : Jz > 0, such that ./F(z) ^> 0 for all z e D. Show that the function is of positive type on D x D (map D conformally on the disk B, and use (a)). (c) Let S be a denumerable infinite subset of the half-plane Jz > 0. Let C(<p) denote the angular sector in this half-plane, defined by [0tz\ ^(^z) cos 9?, for 0 < 9? < JTT. Suppose that there exists a sequence (<jn) of points of S contained in a sector C(<p0) and tending to 0. Let /be a mapping of S into the half-plane «/z > 0. In order that there should exist an analytic function F, defined on the half-plane Jz > 0, such that z) > 0 on this half-plane and such that |F(z)/z| is bounded in every sector C(<p),t follows from Problems 12(e) and ll(c) that A is an eigenvalue